Vinogradov's Conjecture: Least Quadratic Nonresidue

• Vinogradov's Conjecture is a central topic in analytic number theory asserting that the least quadratic nonresidue modulo a prime is O(p^ε) for every fixed ε>0.
• It connects methods such as character sum estimates, sieve theory, and L-function analysis, with strong conditional results under GRH and the Elliott–Halberstam conjecture.
• Recent approaches extend its framework by using combinatorial pattern counting in restricted prime classes, yielding bounds that improve on classical unconditional estimates.

Searching arXiv for recent and relevant papers on “Vinogradov's Conjecture,” especially the least quadratic nonresidue formulation and the terminology overlap with Vinogradov’s mean value theorem. arXiv search query: "Vinogradov's Conjecture least quadratic nonresidue" Vinogradov’s conjecture, in its classical number-theoretic sense, concerns the least quadratic nonresidue modulo a prime. If $n(p)$ denotes the least positive integer that is a quadratic nonresidue modulo the prime $p$, the conjecture asserts that for every fixed $\varepsilon>0$, one has $n(p)=O(p^\varepsilon)$. It lies at the intersection of character sums, sieve theory, and the arithmetic of $L$-functions: it follows from the generalized Riemann hypothesis, is known for almost all primes, and remains open unconditionally in full generality. The modern literature also contains restricted-class results and conditional routes that are stronger in specific regimes, as well as a separate terminological usage in Vinogradov’s mean value theorem that must be distinguished carefully (Tao, 2014).

1. Statement and arithmetic meaning

For a prime $p$, let

$n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$

where $\chi=\left(\frac{\cdot}{p}\right)$ is the quadratic character modulo $p$. Thus $n(p)$ is the least positive integer that fails to be a quadratic residue modulo $p$0 (Tao, 2014).

Vinogradov’s conjecture is the assertion

$p$1

This formulation is the one explicitly identified as “Conjecture 1.1” in the modern literature on the least quadratic nonresidue (Tao, 2014).

The conjecture asks for a power-saving bound that is arbitrarily small in the exponent, uniformly in $p$2. In that sense it is much stronger than classical unconditional estimates coming from character-sum technology, but weaker than the logarithmic bounds known under GRH. Its significance is structural as well as quantitative: a large value of $p$3 means that the quadratic character is identically $p$4 on a long initial interval, so the problem is closely tied to rigidity phenomena for multiplicative characters on short ranges (Tao, 2014).

2. Classical bounds and known implications

The conjecture is known to follow from the generalized Riemann hypothesis. Linnik showed that GRH implies Vinogradov’s conjecture, and Ankeny sharpened this to

$p$5

Thus the conjecture is conditionally settled, with a bound far stronger than $p$6 (Tao, 2014).

Unconditionally, the strongest general classical estimate recorded in the literature cited here is Burgess’s bound

$p$7

This remains the benchmark unconditional result in the general case (Tao, 2014).

There is also an “almost all primes” theorem. For each fixed $p$8, Linnik showed that the number of primes $p$9 with $\varepsilon>0$0 is $\varepsilon>0$1. Thus the conjectural bound fails, if at all, on a very sparse exceptional set (Tao, 2014).

These results place the problem in a characteristic position within analytic number theory. GRH gives very strong control, Burgess gives the best unconditional global theorem, and the remaining gap is precisely the gap between deep but general character-sum bounds and the much more rigid behavior predicted by the conjecture.

3. Conditional and sieve-theoretic approaches

A major conceptual development is Tao’s proof that the Elliott–Halberstam conjecture implies Vinogradov’s conjecture. More precisely, if $\varepsilon>0$2 is a primitive character of prime conductor $\varepsilon>0$3, and $\varepsilon>0$4 is the least $\varepsilon>0$5 with $\varepsilon>0$6, then under Elliott–Halberstam one has

$\varepsilon>0$7

and hence in the quadratic case $\varepsilon>0$8 (Tao, 2014).

The argument begins from the contrapositive assumption that $\varepsilon>0$9 for some fixed $n(p)=O(p^\varepsilon)$0. In the quadratic case this means $n(p)=O(p^\varepsilon)$1 for every $n(p)=O(p^\varepsilon)$2, so a large least nonresidue is converted into a long initial interval on which the character is constant. Tao then studies the logarithmic averages

$n(p)=O(p^\varepsilon)$3

passes to limiting Lipschitz functions, and derives a logarithmic-density form of Wirsing’s equation,

$n(p)=O(p^\varepsilon)$4

This forces the existence of an interval on which the prime-weighted character sum is anomalously large (Tao, 2014).

The second half of the proof converts that anomaly into a contradiction with prime distribution in arithmetic progressions. Using Möbius inversion,

$n(p)=O(p^\varepsilon)$5

and the hypothesis $n(p)=O(p^\varepsilon)$6, Tao shows that $n(p)=O(p^\varepsilon)$7 is supported on $n(p)=O(p^\varepsilon)$8-rough integers. This support restriction makes the convolution sieve-friendly. Elliott–Halberstam is then used to control the relevant arithmetic progression sums, eliminating the large character-sum scenario and proving the conjecture conditionally (Tao, 2014).

This route is important because it is explicitly “non-multiplicative” in flavor: the contradiction comes from additive distribution of primes rather than from zero-free regions or direct $n(p)=O(p^\varepsilon)$9-function methods. Tao also gave variants based on Type II estimates and divisor-function distribution, and noted that an improvement over the Burgess bound would follow from Type II estimates with level of distribution above $L$0, or above $L$1 in the cube-free case (Tao, 2014).

4. Restricted-class pattern counting and “beyond Vinogradov”

A recent and explicitly titled contribution, “Vinogradov’s Conjecture and Beyond,” studies the least quadratic nonresidue through patterns of quadratic residues and nonresidues rather than through character sums alone. Its scope is restricted: it concerns sufficiently large primes

$L$2

and it does not claim an unconditional proof of Vinogradov’s conjecture for all primes (Shivarajkumar, 2021).

The paper considers the residue/nonresidue pattern of consecutive nonzero classes

$L$3

modulo $L$4. Each term is marked $L$5 or $L$6 according to whether it is a quadratic residue or nonresidue, producing one of the $L$7 possible binary patterns of length $L$8 (Shivarajkumar, 2021).

For $L$9, the paper uses the fact that quadratic residues form a skew Hadamard difference set in $p$0, and in particular that

$p$1

This symmetry underlies exact small-length counts and motivates the general pattern-counting theorem (Shivarajkumar, 2021).

Before the threshold regime, the paper records the following small-$p$2 behavior.

Pattern length	Stated occurrence count
$p$3	either $p$4 or $p$5
$p$6	each pattern occurs about $p$7 times

These counts match the heuristic that a prescribed pattern of length $p$8 should occur about $p$9 times (Shivarajkumar, 2021).

The central claim is Theorem 3.5: if $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$0 is sufficiently large, then there exists a constant $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$1 such that the number of occurrences of any prescribed pattern of length $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$2, as $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$3 tends to $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$4, is at most $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$5. The argument is presented in terms of a decomposition of each pattern count into a main term plus an error,

$n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$6

together with combinatorial constraints on overlaps and decompositions of patterns (Shivarajkumar, 2021).

From this, the paper derives Corollary 3.6: $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$7 for sufficiently large primes $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$8. The mechanism is straightforward: if the least nonresidue were large, then the initial segment $n(p):=\min\{n\ge 1:\chi(n)\neq 1\},$9 would all be residues, producing an excessively long all-$\chi=\left(\frac{\cdot}{p}\right)$0 pattern. Bounding the number of long patterns near the threshold $\chi=\left(\frac{\cdot}{p}\right)$1 therefore bounds $\chi=\left(\frac{\cdot}{p}\right)$2 itself (Shivarajkumar, 2021).

The paper describes this as going “beyond Vinogradov’s conjecture,” because $\chi=\left(\frac{\cdot}{p}\right)$3 is much smaller than $\chi=\left(\frac{\cdot}{p}\right)$4 for every fixed $\chi=\left(\frac{\cdot}{p}\right)$5. That description is accurate only in the restricted sense of the strength of the bound on the restricted class $\chi=\left(\frac{\cdot}{p}\right)$6; it is not a proof of the full conjecture for all primes (Shivarajkumar, 2021).

5. Scope, limitations, and common misreadings

The least quadratic nonresidue conjecture remains open in full generality. This point is easy to obscure because some papers obtain stronger bounds on special families of primes or under strong hypotheses. The restricted-class logarithmic claim for $\chi=\left(\frac{\cdot}{p}\right)$7 is one such example: it is stronger than $\chi=\left(\frac{\cdot}{p}\right)$8, but only on that congruence class and only for sufficiently large $\chi=\left(\frac{\cdot}{p}\right)$9 (Shivarajkumar, 2021).

Several limitations are explicit in that literature. The constant $p$0 in the pattern-counting approach is existential rather than explicit, and the phrase “sufficiently large” is not quantified. The argument also uses the special symmetry

$p$1

for $p$2, so it does not assert a corresponding theorem for primes $p$3 (Shivarajkumar, 2021).

A second common misreading is to treat conditional implications as unconditional progress of the same type. Tao’s result shows that Elliott–Halberstam implies Vinogradov’s conjecture, and indeed supplies a conceptually different route to it, but EH itself remains unproved. The result is therefore a conditional implication rather than a resolution of the least nonresidue problem (Tao, 2014).

A third source of confusion is the phrase “beyond Vinogradov.” In the restricted $p$4 setting, a logarithmic bound is indeed stronger than the conjectural $p$5 bound. A plausible implication is that pattern counting near the entropy barrier $p$6 can control least-nonresidue questions more sharply than direct study of $p$7 alone. But this does not alter the status of the original global conjecture, which still asks for a bound uniform over all primes (Shivarajkumar, 2021).

6. Terminological ambiguity with Vinogradov’s mean value theorem

In analytic number theory, the phrase “Vinogradov’s Conjecture” can also refer to the main conjecture in Vinogradov’s mean value theorem rather than to the least quadratic nonresidue problem. In that separate context, the conjecture is the sharp bound

$p$8

where $p$9 is the standard Vinogradov mean value attached to moments of degree-$n(p)$0 Weyl sums (Bourgain et al., 2015).

That VMVT conjecture was proved for higher degrees by Bourgain, Demeter, and Guth via sharp $n(p)$1-decoupling for the moment curve (Bourgain et al., 2015), and Wooley later recovered the full main conjecture by nested efficient congruencing in a more arithmetic framework, extending also to algebraic number fields and function fields (Wooley, 2017). The cubic case had already been isolated and proved separately, and then presented in simplified expository form (Wooley, 2014, Heath-Brown, 2015).

The two usages are historically unrelated in content. One concerns the least quadratic nonresidue and multiplicative characters; the other concerns moments of Weyl sums and translation-invariant Diophantine systems. Because both names appear in the literature, the intended meaning must be inferred from context. In the classical problem surrounding $n(p)$2, however, “Vinogradov’s conjecture” refers to the assertion that the least quadratic nonresidue satisfies $n(p)$3 for every fixed $n(p)$4 (Tao, 2014).